Improved Asymptotic Formulae for Statistical Interpretation Based on Likelihood Ratio Tests

In this work, we attempt to refine the classic asymptotic formulae to describe the probability distribution of likelihood-ratio statistical tests. The idea is to split the probability distribution function into two parts. One part is universal and described by the asymptotic formulae. The other part is case-dependent and is estimated explicitly using a 6-bin model proposed in this work. The latter is similar to performing toy simulations and can therefore predict the discrete structures in the probability distributions. The new asymptotic formulae provide a much better differential description of the test statistics. This improved performance is demonstrated in two toy examples for common likelihood ratio statistics.